Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $H$ be a connected Lie subgroup with Lie algebra $\mathfrak{h}$.
We have that $X \in \mathfrak{h} $ iff $exp(tX) \in H \ \ \ \forall t \in \mathbb{R} $ .
I have seen a proof of this but I don't understand why we need that $\forall t \in \mathbb{R}$. An answer to this question shows that it is equivalent to ask $\forall t \in \mathbb{R}$ or just in $\forall t \in \, ]-t_0,t_0[$ for some $t_0$. Therefore I'm not worried about "for all $t$" versus "for $t$ small enough", but about "for just one $t$" versus "for more than one $t$" (of course $t\neq 0$).
I'd like to see an example in which you have some vector $X \in \mathfrak{g}$ such that $exp(X) \in H$ but $X \not \in \mathfrak{h}$. Is this possible?
Take $G=\Bbb C^\times$, $H=\{1\}$ so $\mathfrak h=\{0\}$ and $X=2\pi\mathbf i\notin\mathfrak h$ but $\exp(X)\in H$.